Density minoration of a strongly non-degenerated random variable (Q1028327)

Consider a \(\mathbb{R}^d\)-valued random variable \(F\) defined on the Wiener space; denote by \(D\) the Malliavin derivative and by \(\delta\) the divergence operator on this space. It is well known that if \(F\) is smooth (in the sense of Malliavin's calculus) and if its Malliavin matrix satisfies some non degeneracy condition, then its law has a density. The aim of this article is to find a lower bound for this density. The main tool is the notion of system of covering vector fields of \(F\); this is a variable \((A_1,\ldots,A_d)\) such that \(D_{A_i}(\phi\circ F)=(\partial_i\phi)\circ F\) for any \(i\) and any smooth \(\phi\). A variable of this type can be found by means of the Malliavin matrix of \(F\), but there are other possible choices. The authors prove that if \(\exp(c\sum_i|\delta(A_i)|^\gamma)\) is integrable for some \(\gamma>1\) and \(c>0\), then the density is bounded below by some \(c_\gamma\,\exp(-c_\gamma\|x\|^{\gamma/(\gamma-1)})\). The proof is based on a radial averaging combined with some Riesz transform estimates.